Steric 5-cubes

5-cube	Steric 5-cube	Stericantic 5-cube
Half 5-cube	Steriruncic 5-cube	Steriruncicantic 5-cube
Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes.

Steric 5-cube

Steric 5-cube
Type	uniform polyteron
Schläfli symbol	t0,3{3,32,1} h4{4,3,3,3 }
Coxeter-Dynkin diagram
4-faces	82
Cells	480
Faces	720
Edges	400
Vertices	80
Vertex figure	{3,3}-t1{3,3} antiprism
Coxeter groups	D5, [32,1,1]
Properties	convex

Alternate names

• Steric penteract, runcinated demipenteract
• Small prismated hemipenteract (siphin) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of

(±1,±1,±1,±1,±3)

with an odd number of plus signs.

Images

orthographic projections

Coxeter plane	B5
Graph
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph
Dihedral symmetry	[4]	[4]

Related polytopes

Dimensional family of steric n-cubes
n	5	6	7	8
[1+,4,3n-2] = [3,3n-3,1]	[1+,4,33] = [3,32,1]	[1+,4,34] = [3,33,1]	[1+,4,35] = [3,34,1]	[1+,4,36] = [3,35,1]
Steric figure
Coxeter	=	=	=	=
Schläfli	h4{4,33}	h4{4,34}	h4{4,35}	h4{4,36}

Stericantic 5-cube

Stericantic 5-cube
Type	uniform polyteron
Schläfli symbol	t0,1,3{3,32,1} h2,4{4,3,3,3 }
Coxeter-Dynkin diagram
4-faces	82
Cells	720
Faces	1840
Edges	1680
Vertices	480
Vertex figure
Coxeter groups	D5, [32,1,1]
Properties	convex

Alternate names

• Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5)

with an odd number of plus signs.

Images

orthographic projections

Coxeter plane	B5
Graph
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph
Dihedral symmetry	[4]	[4]

Steriruncic 5-cube

Steriruncic 5-cube
Type	uniform polyteron
Schläfli symbol	t0,2,3{3,32,1} h3,4{4,3,3,3 }
Coxeter-Dynkin diagram
4-faces	82
Cells	560
Faces	1280
Edges	1120
Vertices	320
Vertex figure
Coxeter groups	D5, [32,1,1]
Properties	convex

Alternate names

• Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5)

with an odd number of plus signs.

Images

orthographic projections

Coxeter plane	B5
Graph
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph
Dihedral symmetry	[4]	[4]

Steriruncicantic 5-cube

Steriruncicantic 5-cube
Type	uniform polyteron
Schläfli symbol	t0,1,2,3{3,32,1} h2,3,4{4,3,3,3 }
Coxeter-Dynkin diagram
4-faces	82
Cells	720
Faces	2080
Edges	2400
Vertices	960
Vertex figure
Coxeter groups	D5, [32,1,1]
Properties	convex

Alternate names

• Great prismated hemipenteract (giphin) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 960 vertices of a steriruncicantic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections

Coxeter plane	B5
Graph
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph
Dihedral symmetry	[4]	[4]

Related polytopes

This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform polytera (uniform 5-polytope) that can be constructed from the D₅ symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.

Further reading

• Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York City: Dover. Retrieved 2022-05-19.
• Coxeter, H. S. M. (1995-05-17). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivić (eds.). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons. ISBN 978-0-471-01003-6. LCCN 94047368. OCLC 632987525. OL 7598569M. Retrieved 2022-05-19.
• Coxeter, H. S. M. (1940-12-01). "Regular and Semi Regular Polytopes I". Mathematische Zeitschrift. 46. Springer Nature: 380–407. doi:10.1007/BF01181449. ISSN 1432-1823. S2CID 186237114. Retrieved 2022-05-19.
• Coxeter, H. S. M. (1985-12-01). "Regular and Semi-Regular Polytopes II". Mathematische Zeitschrift. 188 (4). Springer Nature: 559–591. doi:10.1007/BF01161657. ISSN 1432-1823. S2CID 120429557. Retrieved 2022-05-19.
• Coxeter, H. S. M. (1988-03-01). "Regular and Semi-Regular Polytopes III". Mathematische Zeitschrift. 200 (1). Springer Nature: 3–45. doi:10.1007/BF01161745. ISSN 1432-1823. S2CID 186237142. Retrieved 2022-05-19.
• Johnson, Norman W. (1991). Uniform Polytopes (Unfinished manuscript thesis).
• Johnson, Norman W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD thesis). University of Toronto. Retrieved 2022-05-19.

External links

• Weisstein, Eric W. "Hypercube". MathWorld.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary